Modern plants usually employ grate incineration technologies with "state of the art" flue gas cleaning systems. The waste is tipped into a bunker prior to being fed into the furnace [often after some seperation of materials]. In the furnace the waste is incinerated, producing heat, ash and combustion gasses.
The heat may be recovered in the boiler, producing steam, and then electricity. A plant burning 20,000 tons of waste per annum would expect to produce 11MW of electricity [enough to power around 15,000 homes, if the energy ever gets to the public].
Most of the ash is collected in the ash bunker and then sent on, and is all too often dumped in unsafe landfill or even "reused" in building materials - this "bottom ash" is toxic, as is the "fly ash" from the flue. Indeed the flue gas is even accepted to be highly hazardous by the industry - in the UK this is often sent to landfill, although other European countries cannot legally do this [in fact, they end up shipping it to the UK]. The volume of this waste may be around a tenth of that of the waste that entered, so the volume sent to landfill is less than if the waste went straight to landfill.